A Joint Typicality Approach to Compute–Forward

Abstract

This paper presents a joint typicality framework for encoding and decoding nested linear codes in multiuser networks. This framework provides a new perspective on compute-forward within the context of discrete memoryless networks. In particular, it establishes an achievable rate region for computing a linear combination over a discrete memoryless multiple-access channel (MAC). When specialized to the Gaussian MAC, this rate region recovers and improves upon the lattice-based compute-forward rate region of Nazer and Gastpar, thus providing a unified approach for discrete memoryless and Gaussian networks. Furthermore, our framework provides some valuable insights on establishing the optimal decoding rate region for compute-forward by considering joint decoders, progressing beyond most previous works that consider successive cancellation decoding. Specifically, this work establishes an achievable rate region for simultaneously decoding two linear combinations of nested linear codewords from K senders. Index Terms Linear codes, joint decoding, compute-forward, multiple-access channel, relay networks I. INTRODUCTION In network information theory, random i.i.d. ensembles serve as the foundation for the vast majority of coding theorems and analytical tools. As elegantly demonstrated by the textbook of El Gamal and Kim [1], the core results of this theory can be unified via a few powerful packing and covering lemmas. However, starting from the many-help-one source coding example of Körner and Marton [2], it has been well-known that there are coding theorems that seem to require random linear ensembles, as opposed to random i.i.d. ensembles. Recent efforts have demonstrated that linear and lattice codes can yield new achievable rates for relay networks [3]-[9],

Expanding the Compute-and-Forward Framework: Unequal Powers, Signal Levels, and Multiple Linear Combinations

“…The aim of this paper was to create a unified framework that captures techniques (e.g., unequal power allocation, successive decoding) that are useful in these settings. Follow-up efforts have employed this expanded framework to develop a notion of uplink-downlink duality for integer-forcing [17] as well as investigate compute-and-forward for discrete memoryless networks [18].…”

“…It is sometimes convenient to replace the condition in (18) with the stricter condition that σ 2 succ (H, a 1 ) ≤ σ 2 succ (H, a 2 |A 1 ) ≤ · · · ≤ σ 2 succ (H, a L |A L−1 ) . ♦ Remark 10: For any channel matrix H and power matrix P, there always exists a unimodular matrix A and multiple-access mapping I with permutation π for which Theorem 5 applies.…”

“…For example, the initial motivation behind developing this framework was the authors' exploration of integer-forcing for interference alignment [10]. Subsequently, He et al [17] have used this framework to propose a notion of uplinkdownlink duality for integer-forcing and Lim et al [18] used it to propose a discrete memoryless version of compute-andforward.…”

Towards an algebraic network information theory: Simultaneous joint typicality decoding

“…code ensembles, e.g. the multiple-access channel coding theorem proof in [7] and the hybrid coding scheme in [9], except that we use specialized joint typicality lemmas and a Markov lemma developed specifically for nested linear code ensembles [41,Lemma 12]. Ideally, we would like to upper bound the probability term P{(W n…”

“…This follows directly from the R LMAC evaluation from the proof of Corollary 1 in Appendix A.Remark 5. Let R LMAC,old denote the rate region in[41, Theorem 5] and recall the region R MAC in(15). For a…”